Solving Diff. Equations: Population Growth & Changes

In summary, the conversation discussed two differential equations, one dealing with population growth and the other with population increase and decrease. The first equation involves finding when the population is growing the fastest by setting the second derivative of P with respect to t equal to zero. The second equation involves showing that the population is increasing if 200<P<1000 and decreasing if 0<P<200 by evaluating (dp/dt) at different values of p and considering the solutions to the given inequalities.
  • #1
gordda
20
0
Hello
I am having major difficulties with two Differential equation.
1. If I have the equation P=2000/(1+9e^.06t). where p=population and t=time. How do I work out when the population is growing the fastest? To work this question out do I just make t the subject and diff twice and let it equal to zero?

2. The diff equation dp/dt= .08p(1-p/1000)(1-200/p) has an inital population of Po how do I show that the population is increasing if 200<P<1000 and decreasing if 0<P<200.

Please help

Thanx:)
 
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  • #2
gordda said:
Hello
I am having major difficulties with two Differential equation.
1. If I have the equation P=2000/(1+9e^.06t). where p=population and t=time. How do I work out when the population is growing the fastest? To work this question out do I just make t the subject and diff twice and let it equal to zero?

2. The diff equation dp/dt= .08p(1-p/1000)(1-200/p) has an inital population of Po how do I show that the population is increasing if 200<P<1000 and decreasing if 0<P<200.
1) set d2P/dt2 = 0 and solve for t is a good beginning.
2) show that (dp/dt)>0 when 200<P<1000, and (dp/dt)<0 when 0<P<200. it's not difficult.
 
  • #3
ok i get 1 but in 2 how do i show that dp/dt is >0 or <0.
 
  • #4
gordda said:
ok i get 1 but in 2 how do i show that dp/dt is >0 or <0.
you're thinking it's more difficult than it really is.
begin by evaluating (dp/dt) at p=100 and p=500.
which term changes sign? do any other terms change sign?
now generalize these observations over the 2 given ranges of p.
 
Last edited:
  • #5
shouldn't i use values more closer to the range
 
  • #6
gordda said:
shouldn't i use values more closer to the range
you can.
however, what are the solutions to the 2 inequalities below (for p > 0)?
(1-200/p) < 0
(1-200/p) > 0
and under what condition does the following inequality hold:
(1-p/1000) > 0
the above inequality solutions give the 2 ranges of p.
 
  • #7
geosonel said:
you can.
however, what are the solutions to the 2 inequalities below (for p > 0)?
(1-200/p) < 0
(1-200/p) > 0
and under what condition does the following inequality hold:
(1-p/1000) > 0
the above inequality solutions give the 2 ranges of p.
if you need review of solving inequality equations, try here:
http://www.sosmath.com/algebra/inequalities/ineq01/ineq01.html
proceed page by page thru the various review topics to the extent needed.
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivative. It is used to model various physical, biological, and social phenomena.

2. How are differential equations used to model population growth?

Differential equations are used to model population growth by describing the rate of change of a population over time. This can be done by considering factors such as birth rates, death rates, immigration, and emigration.

3. What is the logistic growth model?

The logistic growth model is a type of differential equation that is commonly used to model population growth. It takes into account factors such as carrying capacity, which is the maximum population size that an environment can sustain.

4. How do you solve a differential equation?

There are various methods for solving differential equations, including separation of variables, substitution, and using integrating factors. The specific method used depends on the type of differential equation and its initial conditions.

5. How do changes in population affect the solution of a differential equation?

Changes in population, such as increases or decreases in birth rates or immigration, can affect the parameters of a differential equation and therefore change the overall solution. For example, a higher birth rate can lead to a faster population growth rate, resulting in a steeper curve on the graph of the solution.

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